The minimum possible value of (x²-6x+10)/(3-x), for x

Question

The minimum possible value of (x²-6x+10)/(3-x), for x<3, is:

UPSC Quizroom · Accepted Answer

Step 1: Complete the square in the numerator

x^2 - 6x + 10 = (x - 3)^2 + 1

So:

f(x) = ((x - 3)^2 + 1) / (3 - x)

Step 2: Substitute y = 3 - x (so y > 0)

Then x - 3 = -y, and (x - 3)^2 = y^2

So:

f(x) = (y^2 + 1) / y = y + 1 / y, for y > 0

Step 3: Minimize f(y) = y + 1 / y

By the AM-GM inequality:

y + 1 / y ≥ 2

Equality occurs when y = 1

Step 4: Find corresponding x

y = 3 - x = 1 → x = 2

Minimum value:

f_min = 2

Answer: 2

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